A Fermi surface changes the weak-attraction pairing problem

Take two added fermions in opposite-momentum, opposite-spin states above the Fermi energy. The occupied sea blocks states below it. Let ξ denote the single-particle energy measured upward from the Fermi energy, and retain an attractive matrix element −G only for 0<ξ<Ec. Use a box-normalized convention in which G has energy units and N₀ is the number of allowed pair labels per unit ξ, equal to the single-spin density of states in this simple model. Then g=N₀G is dimensionless.

Assume this density of states is nearly constant in the retained shell, and seek a pair energy below the two-particle threshold by a positive amount ε. The Schrödinger equation makes each pair amplitude proportional to one over 2ξ+ε. Summing those amplitudes gives a scalar consistency condition. The sea has no dynamics here: no other pairs rearrange, no screening is derived, and no phonon mechanism is computed. The attractive interaction and cutoff are inputs to the model, not consequences of its solution.

As the proposed binding tends to zero, the integral diverges logarithmically because the density of pair states remains nonzero at the threshold. For every positive g the continuum equation therefore has a positive solution. Solving the logarithm gives the exact cutoff-model expression below. At weak coupling it is exponentially small compared with the shell energy. This essential dependence cannot be reproduced by a finite polynomial expansion in g around zero; weak attraction can produce a qualitatively new scale while giving a numerically tiny binding energy.

The constant density of states is central to that conclusion. Near the bottom of a three-dimensional vacuum band the density of states vanishes as the square root of energy, and a binding threshold can remain. In a small finite system, discrete level spacing cuts off the continuum logarithm. A vanishing density of states, a different pairing channel, or a retarded frequency-dependent interaction also changes the calculation. The useful statement is an instability of this Fermi-surface model, not an unrestricted theorem that every arbitrarily weak attractive force binds in every setting.

In BCS theory, many overlapping pair states reorganize together, and occupied states on both sides of the Fermi surface participate in a self-consistent ground state. Its weak-coupling gap has a different exponential from this two-added-particle binding scale when the same g convention is used. Equating those two quantities would miss that collective rearrangement. Likewise a bound pair alone does not guarantee phase rigidity across a sample; phase fluctuations and dimensionality can separate pair formation from the onset of macroscopic superfluid response.

The waves lab provides a limited picture of coherent amplitude addition. Reset and set both controls to phase difference zero and amplitude ratio one; predict a summed sine wave of amplitude two. Reset and set phase difference 180 degrees and ratio one; predict cancellation. This demonstrates that relative signs matter when amplitudes are combined. It does not represent Pauli blocking, an attractive interaction, a pair wavefunction, or the logarithmic energy integral. The actual Cooper result comes from the allowed-state spectrum and the self-consistency equation, rather than from classical interference alone.